theorem Th31:
  A is_line & not r in A implies A c= Plane(A,r) & r in Plane(A,r)
  proof
    assume that
A1: A is_line and
A2: not r in A;
    ex r9 be POINT of S st between r,A,r9 &
      Plane(A,r) = half-plane(A,r) \/ A \/ half-plane(A,r9) by A1,A2,Def10;
    then
A3: A c= half-plane(A,r) \/ A & half-plane(A,r) \/ A c= Plane(A,r)
      by XBOOLE_1:7;
    r in half-plane(A,r) & half-plane(A,r) c= Plane(A,r) by Th20,Th30,A1,A2;
    hence thesis by A3;
  end;
